A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. The diagonal of the rectangle is twice the length of the shortest side of the rectangle. The area of the rectangle is 1,089√ 3 square units. What is the length, in units, of the diameter of the circle?

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The correct answer is 66. It’s given that each vertex of the rectangle lies on the circumference of the circle. Therefore, the length of the diameter of the circle is equal to the length of the diagonal of the rectangle. The diagonal of a rectangle forms a right triangle with the shortest and longest sides of the rectangle, where the shortest side and the longest side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. Let s represent the length, in units, of the shortest side of the rectangle. Since it’s given that the diagonal is twice the length of the shortest side, 2s represents the length, in units, of the diagonal of the rectangle. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b, then a 2 + b 2 = c 2 . Substituting s for a and 2s for c in this equation yields s 2 + b 2 = (2s) 2 , or s 2 + b 2 = 4s 2 . Subtracting s 2 from both sides of this equation yields b 2 = 3s 2 . Taking the positive square root of both sides of this equation yields b = s√ 3 . Therefore, the length, in units, of the rectangle’s longest side is s√ 3 . The area of a rectangle is the product of the length of the shortest side and the length of the longest side. The lengths, in units, of the shortest and longest sides of the rectangle are represented by s and s√ 3 , and it’s given that the area of the rectangle is 1,089√ 3 square units. It follows that 1,089√ 3 = s(s√ 3 ), or 1,089√ 3 = s 2 √ 3 . Dividing both sides of this equation by √ 3 yields 1,089 = s 2 . Taking the positive square root of both sides of this equation yields 33 = s. Since the length, in units, of the diagonal is represented by 2s, it follows that the length, in units, of the diagonal is 2(33), or 66. Since the length of the diameter of the circle is equal to the length of the diagonal of the rectangle, the length, in units, of the diameter of the circle is 66.

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